# Full text of "Nuclear Physics by Enrico Fermi: Chapter 2"

## See other formats

```CHAPTER II . I N TERACTION OF RADIATION WITH MATTER
k. ENERGY LOSS BY CHARGED PARTICLES

1, A charged particle moving through matter loses energy
by electromagnetic interactions which raise electrons of the
matter to excited energy states. If an excited level is in the
continuum of states the electron is Ionised; if not, the electron
Is in an excited bound state. In either case the Increment of
energy is taken from the kinetic energy of the incident particle.
In the following section "ionization" will refer to both degrees
of excitation.

Range - total distance traveled by the particle until its
kinetic energy is 0. Before a formula for the range of a par-
ticle can be derived, the rate of energy loss per unit path must
be calculated. The first such calculation. Is due to Bohr*, and
is essentially classical, I.e., non-quantum mechanical.

2. Bohr Formula. Consider one electron of mass m at a
distance b from the path of an incident particle having charge
Je, mass M and velocity V.

a (electron)

— fo= Impact parameter

mass M
charge £e»-
velocity V

A

FIG-. II. 1

Assume the electron ts free and initially at rest, and
moves so slightly during the collision that the electric field
acting on the electron due to the particle can be calculated at
the initial location of the electron. The last assumption Is
not valid for an Incident particle of velocity comparable to
that acquired by the electron.

We shall calculate first the momentum acquired by an elec-
tron during a collision, and from this find the energy acquired.
As the particle passes, the electrostatic force F changes
direction. By symmetry the Impulse Jf (1 dLt parallel to the

path is zero, since for each position of the particle to the
left of A, yielding a forward contribution to the impulse, there
Is a position at equal distance to the right of A giving an equal
but opposite contribution.

t-J

F, dt

We first estimate

,*e' b

The impulse -L to the path is
the order of magnitude of %± '■

If. = J Fi at ^(electrostatic force )X(time of collision)--^- Ky"

More exact computation: Consider a circular" cylinder
centered on the path and passing through the position of the
electron, Fig. II. 2, Let S be the electrostatic field in-
tensity due to the particle VS. The electric flux Is

Js-ds = 4Tr^e

(independent of V)

* Bohr, Phil, Mag . 24, 10 (1913)', 10, 581 (1915)

2?

2C

Bohr Formula

Ch. II

toy Gauss's theorem. If S x = component of £ _L to the path,
then the flux = f% Zv ktl = "tff^ft. Therefore f^jisf^^S ,

The variation of £1 with time at the electron Is the same
function as would toe found by keeping J-e fixed and observing at
a point moving with velocity V along the cylinder surface.
Therefore

T\ib

T

FIG. II. 2

The Ins pulse Ii."i«, £j.€«-*~ yV = *'■■
momentum acquired by the electron,
electron is then

2_nrL omV 2 b z

where p is ike II . ?

The energy acquired by one

II

The number of collisions per unit path length such that b
lies in the range to to b + db is equal to the number of electrons
per cm, length in the shell bounded by cylinders of radii to and
b + db (Fig. I 1.2). If % = number of electrons per cm3, this Is
2n "ft bdb. The energy lost per cm. to such electrons is

cLE(b) - —

^4-7rfe + ^ ^b

rm V'

II. 4

verge

;m

)

II. 5

The total change in energy to all shells in the range
b min t0 b max lB

The limit b max : The force on an electron as a function of
time will be a pulse occupying a %±mp *%■=■ b/V , non-relativistically ,
It can be shown that if l/£ is much less than the vibration fre-
quency 2/ of an electron In an atom, then the electron absorbs no
energy, i.e., the probability for transition to a higher state Is
small ,

Relativistic ally, the electric field of the incident particle
Is contracted in the direction of motion and S± is increased by

a factor

Vf\

-/3'-

A~%

This "sharpens" the impulse given

the electron; the duration 'V of the pulse of force Is now
approximatelyY^^ifTrgF . The Integral of the Impulse is not

changed, since it depends essentially on the product (field
strength)X(duration) .

Ch. II
b T

Bohr Formula

1/ ^— _. b\|7^3=

29

Tjg^ is chosen so that /^ >7^
range integrated. Thus we may set

b_ =

V

V

(cm)

is valid over the

II. 6

where W is an appropriate average freauency for electrons In
the absorbing material.

Problem: Discuss the statement that for \/ v <T, energy
transfer to electrons Is negligible (or indole of adiabatic
invariance) .

Consider the component of the motion of the electron J- to
the path of the incident particle. Let the coordinate of the
electron be y. y = b + d sin^t. y = jy d cos V t. b»d.

A (energy) = {j component of force )X( velocity ) dt

Since cos 6^1

r Cfb)

A( energy) < f^P^j^J^pq^dt =

Vi

T

I

1

Y 2 -i,M &*+/<''

V

a

as 7^r->»i

The limit b min : (l) Classically, the maximum velocity that
can he imparted to the electron (in head-on collision) is less **~
than 2V.* The energy given cannot exceed 4m(2V) 2 . Therefore
b cannot have values that imply a greater energy transfer per
collision than 2mV- As a function of b, the energy transferred
collision la |^ . Values of b smaller than the solution

coll

oer

This determines b mln as

must be excluded in the integration,

ffotfl

J£

[1.7

'

M i. (2 l + ^ hl f claGSlcal - treatment is valid only if the Coulomb
field oi the incident particle varies negligibly over the atraeii-
3i on ^ ot the quantum mechanical wave packet representing the

* H 11 ? ls ea3 ie 8 "t to see in the rest system of the Incident par-
ticle. Then the electron apoears to collide with something
like a rigid wall.

30

Bohr Formula

Gi-

ll

electron, k Is approximately the do Broglic wavelength of the
electron as seen from the Incident particle. In a coordinate
system In which the incident particle is at rest {this nearly
coincides with the center of mass system , for a heavy Incident
particle), the electron has velocity of about V, aBSUffilng its
orbital velocity is much less than V. The momentum of the
electron in this coordinate system is mtfj/sfiZg* and therefore

&** ~^W • Only values of V>4\ have meaning, and therefore

another criterion for b

mm

is

k

M3I

ritiV

II. 8

The larger of (b Eln ) d and (b mln ) aM should be used in the

integration*. For values of V
and therefore II. 8 whould be used

ere o max ->
Using II ,.;

•in* ( b min)> ( b minL
i in II. 5: *"

II. 9

where V is a suitable average of the oscillation frequencies
of the electrons,

.More precise calculation** leads to the following formula
for heavy particles, I.e., not electrons:

r

(erg cm )
(heavv particles)

11.10

where I is the average ionization potential of the electrons of
the absorber, in ergs. The In term as 9 for 1 mev protons in
NTP air .

3. Electrons . There are two main reasons why 11.10 canr.ot
apply to electrons. (l) The derivation assumes that the Incident
particle Is practically undeflected. But the Incident particle
acquires a transverse component of momentum per collision approx-
imately equal to that given to an electron In the absorber, and
if the incident particle Is an electron, the transverse velocity
corresponding to" this momentum will not be negligible. (2) For
collisions between Identical oarticles exchange phenomena must
be taken into account*--*. Eethe *'# gives the following formula
for energy loss by electrons:

oLx/ lw . V^ t 2I 2 073 a ) s / }

___^ . (electrons)

11,11

where I is the average Ionization potential of the atoms of the
absorber and T = relativistic kinetic energy of the electron.

*Ir. cutting the Integral of II. 4 off at \ ln ^O, we have neglected
a term / bm ' T1 - cL E(b) . This is lust If led In "Lecture Series in Nuclear
Physics", LA 24, Lecture XI, printed edition p. 2?.

** Bethe, Handbuch der Fhysik , p. 519

***Mott, Proc. Rov . Soc. 125 , 222, JK26 , 259 (1929)

Ch. II

Bohr Formula, Electron:

Ari approximation for I ig-J?

(■ergs)

11.12

val U ef; iv f ?fS%g-g b 2! Sl ^*™> ****** *> air, ^ the

** -Ot her particles. For
incident particles of identical
charge moving in like absorbers
-a_E/dx is a function of V bhlv
Therefore if -dE/dx as a func- '
t-ion of enernv is Vatintfi for, say
protons of mass y it can be
tound for some other singly
charged particle B by changing
the energy scale so that the new

Energy, in
e . y .

leu

10°

10 I
108

10 ^
10 10

Electrons in Ai r

., dE in e.v = per

^X__^gZcm^

6

19.5 x
3.67
1.69
1.95
2.47
2.79
3.48

LQ

energy values are K B /M D times
the old. The following table

Cl) de r uS?o 3 r.f TXV n 4w enatoleS ^ lllnS ° Ut thG table f °^
energy correction is ™a p v,,+ J i+\, lerent mass, the above

valuers multiplied brrMf-^si "^ IT™^ l0SS

* Uk/*/- > B faince fy enters the formula.

ergs per ,ram cm^^ S^oted^ Gr l0GB ln thiS table is

where f is the thickness in a- CB r a . Fo- thi*
1S n °i : 'o ain3d ? r ?S ^ E ' 7df b ^ '"ultiolvin

e rg~ c n:
of air,

11.13

tab! e -dE/dx i n
g by the density

PROTONS"

In Air

Energy , _dE Mgy

d^

-cm"

300
47
7.6
2.3

^^^PP^X^^OlSi^^

e. v.

2xl0 6

JL

cv

jT e.v.

.... 300
2x101 47

2xl0« 7.6
2.3
2.3

2x10-
2xlD 10

- ,- -■ -- - — £U£LJ^L

,117X10° 300 ,4x10^ 1200T106 iftQ
.117x107 47 4xloI ' l "~"'- '■"'

.117x108 7.6 4xl0 8 30
.117x10- 2.3 UxlO 9 c 2
.UTxlO 10 g,3|4 xl0 10 I 2

188 lllOl 27,^

10 a 5

10 9 ] .6
ilO

10^1.6

portionaf^c^e^dfn-sTfv ot^fS^^^ l0SS W " , Xact ^ P™~
£" c ^ 2 ^r a given "articK ^ d SJ**** ^2 1^ e ^ POT
on two further fee tors- m K Val 2' But " dE /^ denends

? of an atom does no^ increaS T S^' e ^trons rer atom;
in the formula for (ffi/dS J?l e 1, ?i ? ' ? 'f ths v * 1 £S ht -- thusj?^
appearing in the in - t=t > f heavier elements. (2} I

C the -°'- term ' depends on the absorber (Eqn. 11.12)4*.

Wx&rJt^ tSfs e ^« n . e f£ erl f eilts the original, data is the
... na iron, tni. the energy is estimated. We have derived an

See also Livingston and Sethe , Bev . Kod . Phys . 9, p. 26 5 .

m

Bohr Formula, Range

Ch. II

equation of tne for, -dE/dx = f{l). Integrating, we S et

II .14

X as the range of a particle wxwi * rough approximation

nation ^y teP^formM numerically Fpr a ^ ug_ no ^ elatlvl ti _
we assume f(E ) °Cl/V^l/E. Ihen k • o ' bet ter approxiraa-

cally. More precise consideration showB taai a
tion Is*

% _ w3

U* Ef^V

11.15

jk___,o ».*.< A rourli formula Giving
Empirical range- energy formulas . A roij ^ atmog pheric

range of alpha particles in air at 15

the

oressure is

36

■ft*s.32 (Mev) £ C>vl. ( alph aB in air)

II. 16

IMa is correct to ahout 10% ™^\$J^J^™%™£*te

velocities. The general nature of tne range enei^y

shown in Fig. IT»3-

7 Poi-rization_^ffects^ In the T<

derivaUon^f-IlTlO no ^^^.^f
taken of the Influence on one electron
due to the simultaneous motion of tne
otner electrons near it The elect rons
in a region move so as to diminish the
electric field beyond that region This
partial shielding effect increases with
increase in density of *l*ot*onB. ™>
change in -dE/dx due to tnls effect Is
usually small.**

If the index of refraction, n, Is
net one.'the velocity of light is less than c
example, n~1.5 and velocity^ light^ is^ J/2

FIG. II -3

In water, for
c . If the In-

a, particle produ. -

its experimental discoverer***

8. Nature ^f^e_eouation for -dE/dx. Equation II .10 has

the form

^T^ne^or^^ Livingston and

^,e B afeS e in P defali hy Fermi ^ -,%£&* &\$&, l.C

***Cerenkov , Ehvs. Rev. 3E« 378 lUMJ '
o. 261.

[I

Ch. II

Ionization of a Gas

35

W)

30ANTITAT1VE CURVES IN MDNI'GOME RY,
COSMIC RAY PH VSICS } p 35 a

10 Mc 1
FIG. II. 4

lOOMc 2 ENERGY

The curve BCD gives the l/V dependence. At relativistlc
energies V changes little and CD Is asymptotic to V = c. At
relativistic energies, the log term in (V-/l-f5 2 ) changes, and
increases as V— >e, giving the rise in the curve from to E.
At very 1 ov,' energies (region AB) equation 11.10 "breaks down
because the particle has "velocity comparable to that of the
orbital electrons in the absorber, and the efficiency of energy
exchange Is much lower. The particle itself captures electrons
and spends part of Its time with reduced charge.

9. Ionization of a Kas . If ionization is produced in a gas
the ions may be collected by charged electrodes, and the amount
of charge collected will be proportional to the number of ions
produced. The change of potential of one of the electrodes will
depend on the charge collected (and the external circuit) and
therefore on the number of Ions produced. This voltage pulse may
be amplified linearly and measured quantitatively, a 3 with an
oscillograph. A gas chamber for this purpose Is called an Ion-
ization chamber - " .

+ HIGH VOLTAGE

R '

LINEAR
AMPLIFHES

ZL

OSCILLOGRAPH

FIG-. II. 5

In the arrangement In Fig. II. 5, electrons are collected at
the top plate. A 'negative pulse,. of duration determined by R
and the capacity of the ionization chamber and associated circuit,
is produced at the grid of the linear amplifier.

It turns out that there Is a close proportionality between
number of Ions produced and total energy lost by the incident
particle. For most gases one Ion pair (electron plus ionized
atom) is produced for each 32- 3^ e - v - l03t b y the particle, (see
table or following page) . Although empirically the result Is a

* Reference:; on ionization chambers are: Korff , Electron and
Nuclear Counters (Van Nostrand), Rossi and staub , Ionizat ion
Chambers and Counters (McGraw-Hill) .

34

Scattering

Ch. II

simple proportionality between
number of ions and energy spent ,
the explanation is very compli-
cated. Theoretical prediction
of the average energy per ion
pair involves: (l) calculating
the % of all primary collisions
that lead to removal of an elec-
tron in order to know how much
energy is "wasted" on nor.- ionizing
excitation of the atom; (2) cal-
culating what fraction of energy
carried away by primary ionized electrons is used
secondary ionization. This problem has not been
investigated ,

En e rgy

for one ion pair*

Gas

Energy spent for

one ion palr,e.v

H

33.0

He

27.8

N

35.0

32.3

Kc

27. 4

A

25,4

Kr

22.8

Xe

20.S

in producing
completely

Problem: Design an ionization chamber and snecify the char-
acteristics of an associated linear amplifier so that the
system is suitable for measuring the a energy difference
■"■' (UI) and U 2 34 fijll) a d<

between the U

decays

92 1

a +

90

Th

234.

90

qiV,

-„234

-2S~+

92

234,

92

-^34

tr-

.a +9Q Th

230

The following are among the necessary considerations: £1) loss
of energy of particles while still in the emitting substance,
(2) gas to be used, (3) dimensions and electrostatic capacity
of chamber, (4) gain and frequency response necessarv In the
amplifier, (5) rate of emission of partiales by the emitter.

10. High energy g parti cles lose energy mainly by radiation.
This effect is taken up later, in II, section C, 3.

B. SCATTERING DUE TO A COULOMB FIELD

Scattering due to interation of charged oarticles with the
Coulomb field of nuclei is distinguished from* scattering In
which the incident particle enters a nucleus. Only the former
is treated in this section.

Scattering due to c
.all charged particles In
cloud chamber may have a
angle scattering event,
quently, and their track

/3 —

1, Classical calcu]

ollisions with nuclei is observed for
varying degree. An alpha track in a
single kink, indicating one large
Electrons are scattered much more fre-

s are as shown:

o<

~N

screening of the nuclear
the force on an incident
Coulomb force ^%&yOl %
and J- the charge of the
heavy compared to the oa
system is almost at the
and , the angle of defl

ation for single scattering. If the
charge by nearby electrons is neglected,
particle due to one nucleus is the
, where g is the charge of the nucleus
particle. Assume that the nucleus is
rticle so that the center of mass of the
nucleus. Let b, the impact parameter,
ection be defined in Fig. II. 6, p. 35.

For inverse square forces between particles, classical

Substances , p . 8 1 . '

Ch. II

CHARGE ft'

Scattering

35

NUCI-EUS OF
CHARGE 2

MASS M

VELCC 1TY V

FIG. II. 6
mechanics gives the following formula*

TaN 2=^W*

11.17

This formula is valid at non-relativistic velocities, V<< c.
A relativist ioally correct version of 11.19 for small angles 9
is given in the paragraph containing 11.20.

Exact quantum mechanical calculation gives the same formula
provided the nuclear field is exactly a Coulomb field. Both
classically and quantum mechanically the formula is valid only
If the distance of nearest approach of the particle to the
nucleus Is larger than the nuclear radius .

The cross- section for scattering of the
at an angle 9 in the range 19 is defined to
_L to the Initial path of the particle such
passes through this area It Is deflected tay
Since for given particle and nucleus, b is a
the area corresponding to a given & lies at a
and has magnitude dC% = 2tt bee) db. Substltu
of the corresponding angle 8: do^ = 27tb(0)b
the element of solid angle 27T3ln9d9 to find
unit steradian, and substitute for b(e) Its
11.17 for b. Then the cross- section per uni
is

incident particle
be the total area
that If the particle
an angle 9 in d9 .

function of only-.

tins for c In terms
1 (ej&fi . Divide by
the. cross section per
value from solving
t solid angle at

dcr
cLcu

_. i (e a H 3f

I

4\M V 2 / Si*

&%

11.18
Note that most particles are scattered at small angles.

A relativi stically correct equation for 9 as a function of
b when 9 is small can be derived easily, using the same argu-
ments used to derive II. 3. Since now v:e deal with nuclear charge
2 and incident particle of charge ^-* , we must multipy II. 2 by
the nuclear charge In order to get the transverse Impulse Impart-
ed to the incident particle in the collision. This gives

AJo=^£^ 11.19

v yb

If As la the relativlstic momentum of the incident particle,
angle of deflection Is very nearly Z^Jp/Jp , if &Jp«Jp.

the

e

(e

sma

11)

11.20

If we put fsMV (non- relativi stlc) this becomes identical
to 11.17 when 6 Is small and tan 9/2 a^ 9/2. In these formulae

*See, for example, Lindsay , .Physical Mechanics , p . 76 •

^

36

Multiple Scattering

Ch. II

b is limited to distances from the nucleus within which the nu-
clear charge can "be felt, i.e., has not been screened ay nearby
electrons.

2 . Multiple Scattering. Particles , particularly electrons ,
are deflected many times in passing through a foil of metal.
The net angle of deflection, denoted @ , is the result of a sta-
tistical accumulation of single small scattering events. The de-
tailed theory is complicated . A simplified treatment will be
given here. We assume that no paths are complete loops, as in b
FIG. II. 7. It is plausible and can bo shown** that the values of
to the gausslan lay;, i.e., probability f or © in the

For small

Lty
scattering angles ,

e range d.® is
11.21
® P = Z 0^ 11.22

Dp is the net angle of deflection for p collisions
and the bar means the average for many such traversals of
the absorber.

Since statistically the individual events do not differ,

~6? =&* II. 2

j

FIG-. II. 7

(gf=2TrND

Using II .20_Tor 0,
averaging over Q 2 for
values of b from b n j_ n to
bmaxj an< 3- summing for all
collisions in the length
of path U, we obtain ***»

W )uh

11.25

of the foil.

For thin foils, D differs little from the thickB.es
N — no. of atovns uer cc.

Due to screening of the nuclear charge by electrons, the 2
felt by a particle depends on b, therefore Z in the integral is
a function of b. We take 2 outside and adjust for the error by
choice of b^. Assuming the absorber is so thin that _Jp and V
do not change ,

Choice of limits: b max : The equation 11.26 would be
strictly correct if at distances beyond b^ax the screening of
the nucleus were perfect; i.e., no scattering, and for distances
within b m ax there were no screening at all, i.e., full value of
2 uere felt. No such boundary exists, since screening in-
creases gradually with distance, but fortunately the log term

* E. J. Williams, Proc . Roy . Soc . A169 531 (1939); Rossi and

Greisen, Rev . Mod . Plrys . 13 249 (1941),
** Rossi and Greisen, I.e.
*** This is shown in Appendix II ,1
«*■»* Shown in greater detail in Appendix 11,2

Cl1 ' T1 Multiple Scattering

is not sensitive and we mav cut*

gS

V*

37

11.27

me factor ? a takes into account the variation of the function
radius/f ) e4€ " Z ~ (electron charge within sphere of

the maximum angle in a single scattering
process. Since we are not "counting any
values of © > i ,we may impose the rough
restriction that £?<1. This glims

--U&HT ATOMS

HEAVy ATOMS

2^e !

FIG-. II. 8

!*>« V-^o

(from 11.20)

11.28

Otner considerations may govern b^ such as (l), the finite
size of the nucleus: b mln >A / 3 1 . 5 x 10 ~13 j frora ePuatlOT] 1A
? G L° ;; UJ ' 3ize of _™ye packets of particles in tbe colli-
f 3 ° n **- Tne result for @* using our choices for b min and. b^

Jn

a Vyb

11.29

The result Is not sensitive to the choice of b mln and b max ; the
log terra is of the order of magnitude of 10. JuTae to thf°|« fac-
tor, scattering increases rapidly with weight of the elements in
the absorber, even for equal atomic densities. n

tivi^^^S^^C f, X f kinetlc energy). In the extreme rela-
tivists range \jjp*^ kinet i c ene rgy . The re f o re , roughly ,

w

KINETIC ENERGr

11.30

This formula gives the :nost important characteristics of 11.29
Evidently a proton and an electron of the same energy wil n be
ff^ ed a °? Ut the . sanie amount. "But the range of the electron
is ,uch greater, ana therefore, experimentally , large anrle
scattering of light particles is more prominent than of heavy
particles. . ' J

Nuraerical formulae:

ELECTRONS

&'

£*io e D

(KILOe.v.) 2

11.31

ladiaT^^f' 311 S'^aS 1 MeV el « ctron ^ delated about one

Electrons

iw MK (B) 2

j 70oo D
'(kilo e.v.)'

11.32

This is the radius of an atom of charge? using the Thomas
Fermi model; cf. Lindsay, Physi cal Stat i st 1 c a ■ ^ . 226

tK^ ar V ls ?" s ^ in T.dlliaras, I.e., and rn^re briefly, in
J&nossy, Cosmic Rays , para, 173. J '

38

Passage of Electromagnetic Radiation, through Matter Ch.II

scattering in the mind,
than the thickness of t\

Lnd absorption must be interpreted with
The true oath lenght mSv be much longer
le absorber. The equations for range of

particles applied to a homogeneous beam of non- scattered parti-
cles passing through an absorber would yield a curve of number
of particles emerging vs. thickness like FIG. II. 9a. But ex-
perimentally, FIG. II. 9 b ^s more nearly correct for electrons,
and the reason for the difference is straggling due to scatter-
ing;, f NO. EMERGING

f

NO. IN

ORIGINAL

-^ RANGE -

THICKNE5S

FIG. II.

There is a problem on multiple scattering in Appendix II. 3

C. Passag e of Electromagnetic Radiation through Matter

The]
;d Oj

ill

There are three processes by which photons are either ab-
sorbed or lose energy:

Photoelectric absorption
Compton effect
(3) Pair formation.

1. Pho toelectric a bsorption. The electrons bound to an
atom may be grouped into shells according to their binding
energies. The Innermost shell, the K shell, contains the two
Is electrons and is bound with an energy of about Ry (Z-l) 1 -,
where Ry is the Rydberg, about 13-5 e.v. The next shells. are
the L and M shells. Electrons in the L and M shells are bound
With energies roughly Ry (2-5 ) 2 A and Ry (Z-13) 2 /9, respectively.
These last two numbers are very approximate, since the energy de-
pends on screening and the shape of the orbit. The last filled
orbits of the atom are the valence orbits producing optical
spectra. Beyond this, the states are unfilled. Provided its
energy is sufficient, a photon may remove an electron from any
of the shells and leave it either in one of the previously un-
occupied bound states, or in an ionized state. As the frequency
p of Incident photons is increased from zero, they first become
able to excite electrons in optical orbits of a feu e.v, binding
energy. This loss of e-ergy is the first contribution to absorp-
tion of photons by photoelectric effect. Asj/ is further increas-
ed, electrons in deeper lying orbits may absorb energy. When V
reaches the binding energy' of "a particular shell of electrons,
there is 'a sharp rise in absorption. A rough plot of the ab-
sorption as a function of V is given in FIG. 11.10.

The point
ab so rot ion

Lt whl eft th e a h so rp t i on change s abrupt 1 y du e to,
sav, absorption by K electrons is called the K absorption edge,
and is at the energy at which K electrons are lifted to energy
levels > 0. Energy is the energy
zero velocity.

of an ionized electron with

sub

The L absorpti
■shells 25, 2Pk

or.

edges are three-fold, since the L shell has
md 2Pa/ having different energies. This

Oil. II

Photoelectric Absorption

39

energy splitting, is due to (1) different screening of the nucleus
for S ana P orbits, and (2) electron spin-orbit interaction split-
ting of Py and Py .

-Z-
Q
fc-

Q-
CC
O
u~>
CO
<

K EDGE

DECREASE -AS p>

^B.ET. f k ORBIT,
FIG. 11.10

■ 35 KILO e.v -for TIN Z=5Q

Tne cross-section for photoelectric absorption by one K
shell electron (for transitions to the continuum of energy >0)
has been calculated by quantum mechanics, assuming a hydrog<=n-
like wave function for the electron (Is) . The result is*

(3 (jW) ^er tatlectvon =

I28TT e'

3 e -4scot-'e

m& zy+ |— q-

•zve

11.33

where Z^

.. is the frequency of the K absorption edge, 1/ is the
frequency of the photon and is greater than l£ , and £=\Tj^

The following simplified formula holds for V near to t^ :

&fe

(5 p (tV) jaer Is electron «~

S.3:' 'O'^JZJ^

z z

Mtteir to 1A.

11.34

The mass absorption coefficient A is defined hy the relation
(fraction of photons not absorbed when passing through thickness
§ m g-cm tJ of absorber )se4>k*f). A s i m ii ar equation defines the
absorption coef f icient x . i.e., the fraction massing through
thicxness x m cm is equal to ~-xx
" fc;

Problem: Derive the relation between >u and cr per atom Cal-

Consider a beam of/n, photons-am -2 - sec -1 impinging on a sec-
tion of absorber of thickness dx and cross-sectional area S
An area B given hy B = V ( cm^ ) X N (cm" S ) X S (cm 2 ) X dx (cm)
may be considered to absorb all photons hitting it. B£s.
The number of photons absorbed per second is (B/s)^ , The

emerging team has density^ per cm 2 per sec'

g <ffa

= -dHcbx

sk~sw e

—trMa.

A STL

The fraction net absorbed in a finite distance x (cm

^ = P^ N ^ = ^^f"

m

= C

is
by definition.

Heitler, quantum Theory of Radiation , 2nd ed/p. 124

■■-■ Photoelectric Absorption cii. II

But r*»y where /> is the density in g-cm-3. Therefore

^ = f(^Y)"-^ m W™ ^ N = £* ,o 23

11.35

£^ f b * °^culated by equation 11.33, multiplied by the num-
ber of K electrons P er atom, i.e., £, an d^ found by II 35
Tills calculation ignores the contribution^! the L^M N etc'
lit) ?h, f C r?i lnS t0 Rutherford . Chadwick & Elli . ;,

4 n ^ v S ? e ii v aCC T ° T U ? tS ** -a* of absorption for photons
ene?Sy 5 19 E 3 ^ ^ S e ^i° n 11.34,^/ for lead at photon
energy- .19 mc la -.,-6 en--g X, Actually,* is— 8 In
1 x nr, 1 ng^ , 5 . 76 , we i gno red ab o r ?t i o n in other than K shell.

tth change "of K

The sharpness of the absorption edges nohes pee
methcu of measuring the frequency of ^ rays of low e
J edges yary for different elements, and one may det
tween which two K edges the unknown nhoton erern; 11
lng the sudden change in absorption ~ '
of the absorber used.

sible

■■ =1 LiJ ■

ermine
e s by-
edge

The

e-
ohse re-
location

a*

C

u

S

n i

D l

\\

*

1

\

1
\

.Oc

>:5

.0

2

.&

\
7 .

9

rNERGY IN UNITS OF "mc*

LOCATION OF K ABSORPTION EDGES
PIG. 11.11

Proolem: Giver, a source of 05 kilovolt H radiation. What
absorbers would you use to bracket this energy and thus~ serve
to determine it? Determine the thickness to be used. Qlve
r™n Q8 l0n t0 thS Pliability of the chosen absorbers.
Compounds may oe used. The absorption of a compound is the
sum of the separate absorptions due to the constituent
element s.

3 .
'-&V in.

Comoto n scattering . Photoelectric ab so rot ion decreases
r-eases. Soon after the K edge is massed the oredomlnant
process removing energy fro.-, a beam is scattering by electrons
Si iV?\ true abso ^Ption, since most of the energy Is ro?
absorbed but rather sent in a different direction"? In the low
energy region h V « m c 2 , scattering cross-section aln be cal-
culated classically and is given by the Thomson foxmSia:

elec+ror^Vt^i^^^f 12 ^ ^"t^nagMtt* wave impinges on an
AsluS The JtJ leCt r n / 3 SSt ln m ° ti0n and therefore radiates.
,^T J*f el ^ctron is free, a good approximation If ^ of the
is tha ^rT h h^G^ than the frequency of the electron I? A
is the acceleration of the electron, "

e£ =rmA

PLANE POLARIZED

A^

A =

PHOTOH

K

■ar L

FIG. 11,12

"" f ELECTION

ELECTRIC VECTQK £
OF £, M. WAVE.

Ch. II

Compton Scattering

*L

The instantaneous power radiated from a moving electron is*
dt " 3 c3 11.36

2 e P T

3 c 4 /w>*

In this case the average rower ia j£

ar

This energy taken from the incident wave and re- radiated may-
be expressed in terms of a cross- section C% per electron defined
so that

Oj X (Intensity of Incident beam)= un it time by the electron)

Q^cra 2 ) X I (erg-cm^-sec"- 1 ') = yy (erg-sec^ 1 ) 11.37

This definition is equivalent to saying that all the energy in-
cident upon a surface Cfr i s absorbed and re?- radiated. The
intensity of the incident beam is

I --T^sz. (evtf-o^-sec ')

Therefore

U T — r _ lM = T' r e' = c - 66 x 10 "• -'(cm ) ( per electron ) II. 38

where r g Is the classical radius of the electron, 2 .8 x lO" 1 ^ em.

The Thomson cross- section is evidently independent of fre-
quency. 11.38 breaks down when h.p is near or above mc =^^Mev.
Then the phenomenon must be treated as Compton scattering. In
the Compton effect, an incident photon \-JJ collides with an
electron. Compared with the energy of Irz* , the binding energy
of the electron is negligible. The scattered photon suffers a
change of energy which can be calculated by imposing conservation
of relativistic energy and relativistic momentum on the collision**.
The result is

■"Scattered — A incident = X^T \\ -Cos ©J

11-39

nV Incident

Vixj sc&ti&red

electron

The frequency of the scattered photon is less than that of the
primary* The factor h/mc«^ c is the Cometon wavelength,
0.024 x ID -8 cm.

The cross-section for Compton scattering Is given by the
Klein-Nishina formula***:

11.40

^Abraham- Becker, Tbeorie der Electrizitfit , Vol. II. p. 73
**Thi's is done, for example, in Rlehtmeyer and Kennard, Intro -
duction to Modern Physics, 3rd ed . p.. 533.
*'«*Derived in Heitler, I.e., p. 149

42

■

Compton Scattering

Ch. II

Problem: F ind asymptotic Bxnreualora for II 4n nZ i ^

enemies and for higJl energle5 _ ^ ^11 .40 'rouS^! "

The asymptotic expressions are:

.Low .energies: C = a T [ 1 -Jtw- + ^**+ ... j , <*«! H.4I

HiSh energies: eg « fc^ka*-^) rt>>1

The Klein-Nishina formula is plotted below;*

\AZ

Direction of

POLARIZATION

rorJa n ^I?3 r 8 d L1Sf1L 0n a„4^ r / d T st en ?^i r where the Thomson
13 that due to a radiating c^atlica rtrll «* 1 « a ° f *»**tad ener^

ft'VS ^ ^ the an S le between

iBBrit S 4 ■ - Primary radiation

azW* P f 1 ^' an ai ™3* over the

5ffi£*}* ^* tne maxima of scattered

CP J^p " hen ^f approaches and ex-

'INCIDENT

FIG-.

¥+9^30°

11.13

lm,enSt e y S ifS foTow'nJ ■" ""S^ ai ***»M*» of scattered

Jl a | +^(1-^.9) J

H.43

Sw/J Tn?I * thS 1 J?S 1 ? f* Ate the direction 9^ ■ a
cident radlatlonT^, Sein \i2^f ^ <* P laM polarized' 1?-

from integrating this eqiaSon! ^ II,4 ° WOuId res ^

* From Janossy, p 05mlc Rays ( (0xford)i

Ch. II

43

3 ■ Radiation loss by fast el ec trons {ore m s st rah lung ) .
This paragraph should coin© under II " XJ sub-section 10, butls
placed here be cause our calculation makes use of the Thomson
formula for OV , derived in the last sub- section.

Consider an electron of velocity V ^=c -passing a nucleus 2e,
We calculate the energy given off by the electron in the forn of
photons when it is accelerated by the field of the nucleus.

FAST ELECTRON

In the rest system of the electron, the nucleus Ze moves with
velocity almost c. The electric field £ of the nucleus is con-
tracted, and the associated magnetic field H is X. to £ and of
about the same magnitude. Therefore the moving nucleus looks to
the electron like a plane electromagnetic wave. This will be
discussed more fully shortly. The wave of photons representing
the nucleus suffers Compton scattering by the electron. These
scattered photons, when viewed from the rest system of the nucleus
(the laboratory system) appear as the photons emitted by the in-
cident electron. According to this brief summary, the calcula-
tion has the following parts:

(a) Lorentz transform to the rest system of the electron.

(b) Calculate- the characteristics of the Dulse of electro-
magnetic radiation that represents the moving nucleus.

(c) Find the density of photons corresconding to the pulse

(d) Calculate the probability of scattering of these photons
by the electron.

(e) Transform back to the laboratory frame.

Throughout this calculation we leave out numerical factors.
Their inclusion would obscure the Ideas involved. The correct
numerical factor is Inserted at the end.

(a) Let p = %«1 I i* IA/I-/3*' is large
the electron in the laboratory frame is Ymc^.

The energy of

LAB, FRAME'

ELECTRON
FRAME

Je

FIG. II.15

44

Ch. II

In the lac frame, denoted by °, the electric field of the
nucleus is \£°| = %*/&. In the electron frame, denoted *, Its
perpendicular component £* is Increased by the factor H\

The magnetic f ield °&* seen by the electron in its own rest
system i s _L to both £* and the velocity vector. Denote this
field by ^ = | -A*\ . Then* Kj = € L %0 *^S ^ *| . V ,
Therefore y* = e * ( as ln a plaa@ electromagnetic wave.

(b) The reel on in which the electromagnetic field is not
zero has a length, In the electron frame, of about b/v , the
l actor g coming from the Lorentz contraction

SHAPF OFPUL5E: SPECTRAL DISTRIBUTION;

t-0 when the electron is in wave- front Diane representing
■nucleus " °

FIG. 11.16

The spectral analysis of this pulse raay he done by assuming
for simplicity that it is a Gauss! sn of time width b/fc,;

e, -x, cc e 2be Z1 . 44

SSto^iSSli^? 11011 ° f C °^° nent ****»*« l- -then

di ^Lsrtr.WiL^^ gst^Tr p 2 e ^^sr ic

volume is dven bir S +X ari p- 3si I ?cY r _,

n rrn ™p f^f x Y? n D/ Tjr;" p" V*F*7 ' , . Tae volume per unit

area of the disc ecuals the thickness = b/y (cm Therefore the

energy oer unit area Is ° ' re 1 ' ne

(c) Approximate to the gaussian spectral distribution by a
rectangle of width Xc/b ***■• This means that the energy carried
by photons In the frequency range Ay* is simolv prooortkmal to

*5h raham-Eecker, Vol. II, p. 48. " ~~ ~

**Strattoh, Electromagnetic Theory , p. 290.

»**it will turn out later, (just before 11.54) that only frequen-
cies in the electron system less than p* = mc^/h will be inte-
grated over, and that tyj* » S**g .<< '|^fc , Therefore the portion
of the rectangle used will be at a relatively flat oart of the
par s si an , a s sho wn i n F I g . 1 1 . 1 7 .

Ch - II Radiation by Fast Electrons

<-o

V.

MSI.

^C

therefore the energy carnled by
photons d: this rr n - :e a*,* ^

m a at distance b, i§ P

Az--*b ? 2 e^V

The number of photons tier ~t 2 in
the range A^ is equal to this
XCenergj per photon^ 1 which is^)" 1
Therefore the number of oho tons

£X + m l;V h ? ^anoncy" range

T.r_oe bis giver, bv ;

A^*at the dial

FIG- 11.17

S&hS^'b

b 3 ^c hv"

o V y* > &

b

fj. i ™ f W tor ^ > * —

I aj The mean number of o,„ t , Q „, ^ D

frequency ^* occurri fi|i? the S SKTl" *** ?hotons of

GQ X (density (per cnT 2 ) o
Therefore

No. of scattering everts

photons of free. V* )

for rre a< ft~^ ^Wi^L *V A£* , «

01 h, am for one nucleus / ^h b a ^* c ^ J

II .45

offfl^!S cross-section per nucleus be £fe*)*ft/*- ■ *

whi^ Beyond b max ^X/^* ths f T&QV[miir y ,* (Y

which we are commit 1 -ir- vr TJ *\ \ * ■. ^-^^.„/ ^ (for

is the case and use max > W' - e assume this

u min

'max
is subject to seve:

limitations.

II .47

Sends on the d-.-m-i n,,-i~~-.,~ "~ """"'" Ul " L,J - u:!B ' Which is dominant de-
must bel Lt " ^ The ni0st ^Portant is that b ,
ing the ^«1^S^;^^ n ^5«±«»« ^ Packet' reor^e^t-
the Compton wavelength i * wl le ? S , h for the electron is
velocity c/^ ,ithre S ^t to +X ^ *** *** electron have
4e Broglle vlvelength £ *° d *??,?** S ^ m > 1*1 a giving it

velocities. Taking K^_ - f
>-- 1 "mm - A

Prna »,
tarn,-.

a

',et:

4 = 137H"'

11,43

tm s pomt is w«asxs sjfgs/itif^^: 1 ^^^^-.

m

More nreciso

acuiation leads to 133 Z' for II. 45.

^v*)av*^^*<1 c JU

173
§1?

5,0

SVSt«

Fro:

( e ) T r a n s f o m. ":ac]i to the lab
the photon changes from ~p* to ~i/ a

Therefore

Introducing this result into 11.49,

'-Kh=-nr^e

Ch. II
Therefore
II. ho

The frequency of
special relativity*

11.50

11.51

et;

11.52

The factor 4 /%. results from exact derivation. The cross- section
per nucleus for scattering at frequency J/ in the lab system is

^•JA^Z^/A-H

11.53

where e~/c)i, the fine- structure constant, has been renlaced by
1/137 ■

Before integrating to find the total energy loss by the
e 1 e c t ron , we e xam Ine the limits of validity of thi s re suit , In
11.49 the maximum value for Oc was used. This breaks down at

[IV* = mc 2 , This defines the limiting value of I/* , namely Z^" .
Corresponding to 7-i* are two values of ~U° , one for forward and
one for backward scattering in the electron frame. (Thomson
sc at t e r I ng 1 s symme t r i c al , f on;?, rd and ba c k . } The 1 a rge r value
of ~P a results for a numerator in 11.50 of about two , i.e.,

Ke, = H

* l+/3cr>vB*

m

Therefore the use of the Thomson scattering formula 11.38 causes
the theory to break down wher- h^°/V = mc'-, or

But thi s sav s 1/

:i!;.X

enenrv of ele ctron

11.54

, which clearly must be

so , because the electron cannot radiate a photon of energy great-
er than its Initial energy.

The expression 11.53 diverges at low frequencies. But the
enerrv carried by low frequency photons is finite. The total

TJ'

per oath lenr.tb dx is

energy loss to photon^ of freriuenc;.
equal to :

(energy ^>er |oinoton) X (no. of nucleytm^X (crass - section "£{V)AV") X &X

Therefore -(cIeJrao. , the total average energy loss per path length
dx for all frequencies Is given by:

ft 4 ". JzjVsO

11.55

* cf. Abraham-Eecker, Vol. II, u, 312.

Ch. II

Pair Format i

4.7

^-^^f^

II

'.■.'".lore

!

nzC = energy of s ].e<

tron,

ete^fT^t/lf^ 71 ^ '~ Xve an a FP re ^bl e amount of its
rro^^e^oraSo^ffif^'l^? ^ 1 ***** l0M ** *** Wid ^

(fiE7dS) rad is evidently proportional to energy. It Is the re -
hi 3 encr^ief *" C ° SmlC ra7 P h 7 3ics w ^e particles have very

We may v/rite 11,56 in tenai

Define
Then

X-

-m-^^Y*

(3e) = _ e ^c

x -vat, F- s

RID

or

°- ^ , the ra diation length ,

11.57

jar

For 3f«.4 , I/2.7-

c

-£

Roughly ,

rn

E^E^e -" 11.58

the energy Is dissipated, on tho average
whs re C is nearly constant. 11.59

Jk «C

Radiation loss r>er g-civT 2 traversed
is greater for heavy elements. The
ratio of radiative energy loss to
Ionization loss, for electrons, is
given rouahlv by*

^Ens

Material 1 n lEnergy for

. ccsjiton ~ djxjiot-i.

, Air j NTP
J Alurni:iu:i

"330 rn:

9,7 cm,

.517 cm.

ISO Mev
5S
7.0

? * ( Me vj
800

H.6O

(Thickness of the atmos-
phere is equivalent to 8
fen. of WTP air, or about
SOJJt )

+ jor , Js e r^ rotation . We return to the discussion of absorp-
tion 01 electromagnetic radiation in matter. The mos+ limoS

S^i^S Ve T 7 r- lGh ener ^ ^ectroma^tir^atWirby
SSnKSi ?" tMS PrCGBSS a ph ° ton diaappeara and a ros-
Dir°c's ^;\ n 4ifi r ° n appear * This is t0 be underatood only by
is slrr^ii^L ? 1C ™y e /«^hanics. The following discussion
is aimplliied and qualitative**.

According to the relatlyJ^Mctheory of the election an
electron tea energy + xfu^WTZX^ mJ. \. Le ^~ -' ^ n

lj -yinifi ) + p c . This equation permits

"This appears .inverted, in Lecture XIII of Los Alamos Reoort #24
d\lt ?£ prin ~ Gd version. Also inverted in page 72 of fia'setti.
Trie theory is given, for examole, in Heitler, I c

48

Pair Formation

Oh. II

I

negative energy values. The energy spectrum of a free electron
looks like Fig. 11,16.

In Dirac' s theory, practically all
negative states are filled at all points
in space. A vacuum is then a sea of
electrons in negative energy states. The

presence of this charge is rot observed +<mc :

because it is uniformly distributed.

A photon of sufficiently high energy

■-

I
i

FIG. II

18

may lift an electron from a negative
energy state to a positive energy state
The energy threshold for the photon is
2mc 2 , since for a free electron there
are no states between -mc 2 and +nic 2 .
Physically, this means that the photon
must supply enough energy to create two
particles of mass m. Momentum must be
conserved and this requires either that
the negative energy electron be near a
nucleus or an electron, i.e., not free, or that two ohotons
coming from different directions coalesce and lift an electron
from a negative energy state*. If the electron is near a nucleus
it may occupy discrete states just below +mc 2 . These are within
a few e.v. of 510,000 e.v. Strictly, then, the threshold for
pair formation near a nucleus is 2mc 2 - (binding energy of
electron). This is of no importance because binding energy «mc 2
and because transitions from negative energy states to the dis-
crete part of the spectrum are improbable and not yet observed.

The result for cross- section for uair oroduction near a
nucleus Is calculated in Heitler** and is

II.61

In the extreme relativistic rai

en

rmr is independent of energy,

For other energies the situation is more complicated, and is
described in Heitler.

(A simplified procedure similar to that used to comnute the
radiation loss by fast electrons may be used to comoute the cross-
section for pair formation at very high energies, if the orocess
of collision of photons in a vacuum {h v + hv-*e + + e~J is known
This is outlined. Consider a uhoton of \iv ^ 1000 mc 2 or so.
Transform to a new coordinate system in which the nucleus moves
very fast, almost, but not quite, c. The ohoton seen in the new
system has reduced frequency. The nucleus looks like a wave of
photons '(just as in subsection 3). Oive the nucleus such a vel-
ocity that its photons have the same (reduced) frequency as the
incident photon seen in the new coordinate system. The" process
i ■:-■, the simpler one of collision of ohoton s in a vacuum, '

h v

W-

+ e"

)

shown in Appendix II. 4 that momentum cannot be conserved

_r f OTVPra.t, 4 r nn Vnr r->no 1 onl a+ aA v-o, ,-,+ ,-. .«

Xt is onunu j_n .tiuufciiJCLx j.1. ^f t,nat. momenta
in pair formation by one isolated ohoton,
*#Heitl g;r, i.e., p. 200

Ch.. II Pair Formation

We may define a mean free path for pair production

49

Jp^

I

II .61

Then the decrease in intensityynof a beam of photons will follow
the equations:

7C

dm b- „ £* /jt =^ Q e ^

11.63

/n

^

4-

i

23 -Z a N _2 S 183
5 T37 yCe ^-^

11.64

This is analogous to the equations for radiation loss by an
electron, 11.58 and 11.57

e -i (H.58)

E

J*

&

Evidently

4m z H„a j U3.

-*-•£-<

(11.57)

11.65

COSMIC KAY

5. The phenomena of radiation and pair formation are
responsible for the shower phenomenon in cosmic rays*. The
shower is initiated by an electron
haying energy of the order of
billions of c.v. By radiation it
produces photons of comparable
energy. These photons produce
electron-positron pairs, each
energy is soon divided among many
particles. When the energy of a
particle reaches the point where
ionization loss predominates, the.
shower stons.

Me sons have ma s s about
The factor (e 2 /mc2)2 appears
meson, this factor is(l/2.Q0r

00 m.
in the
For a
as

large as for electrons, hence energy
loss by. the meson by this process is
much smaller. A meson looses energy mainly
a meson of some billions of e.v. ionization
the meson can penetrate several meters of 1

FIG. 11.19

by ionization.
loss is small,

For
and

b. Summary . The phenomena of absorption of photons can be
summarized in the following graphs.

* The theory of showers is given in an article by Rossi and
Greisen, Rev . Mod . Phys . 13_, 249 (1941 )

' D ° Absorption of Photons

GENERAL FEATURES OF ABSORPTION AS A FUNCTION OF ENERGY:

Ch. II

PHOTOELECTRIC
ABSOF^PTIOW

PAIR FORMATION _

.01

COMPTON EFFECT
f (KLEIN -N1SMINA)

' l0 ENERGY^nric' 1 UHTO 100

FIG. II. 20
ABSORPTION COEFFICIENT FQ* PHOTONS iN LEAD, AS A FUNCTION OF ENE^T: *

h
z

Ur

U-
U
O

o

z
5

Q.

B

8
8

\

Kw - limits nrrLC 1

FlS.II. 2.1

MATERIAL

Alur

Colpf3ey

Location of tninimum in
total obsor^tion

-35 MEV

~IO

3M*

Thickness F j in 5/01' 1 f-w l f&f*\$Q
dedu ct mn an tntemi-iy

■ 47 &£m 3 at -35MW

•31

-10 Mei

-15

-3J£m<

Problem: Suppose e ^gos-tUg ray electron of energy ld00~lFc 2 ~

tut the electron lec.ves the plate ^conroanled by a nair whose
total energy (electron plus positron) is larger than^OO m^ 7
assume tne pr.rtic_Les chan-e direction nenlifri t>ly .

Use 11.53, wri+lnc It in terms Q f^ ■ grf'^ !

To ? ;et the cross- section for all the nuclei ^thickness dx,
by U dx, -ettlnr

multiol:

feU ^

* From Reltlt

216,

Ch. II

Aftiendix H.l

cro

53 S9 ; u ^Sj hotons ir the ^ 5o ° - 2 *> 1000 BQ *

J WjT -

A ^°_ .693 j
~^77 - — ^— OA"

Will

This is the nrobabillti tha j - t. wu j

be produced In LJ tSkSX dx^ " ° f ener * r > *

The probability that a photon ,^

duced at X will then form a ,air

fron 11.63. The pair Produced
Will nave all the ener 3 of the

that both- events will occur Is

[using H.65 For lead in this energy region

[^^ Ihereiore the probability is O.OO75

the scattering anrl e fn-n - =■ i " "■■ " > Where &. j S

over .any traveraSs 0? the ^ftrbe^Tt^ ^ ^ lc ***«>
alated angle between the IniUal df^.ff^ ^ b9 tKe net a ^uin-
particle's velocity vector \^ ?£ Q ]J?^™* ^d the

the traversal in ah&<?ti <-m a lii + re , " L1 . e -^.+ i collision o^
next paR e. W * atwm ' A Sorter derivation appears on the

NOT IN PLANE OF
PAPER IF i. ^o

TRIANGLE 5

4+1

JM COLLISION-)

A r -'V= ,to -^ + - l

I^TIAL DirecT.ONTOWH.CH © k IS MEASURED

FIC-. 11.22
This is a recursion^^I 2 ;, L d :^—^ *** respect to f,

II. 67

ii™ S^Jiri, 8 * P^ 1 ' '"'- "^- «- -"•■

'" #•- #(.-£?«

ST

£ ? S j~ P**4 gj»- a- jf |s

52

Appendix II .3

Ch. II

Shorter derivaxion of 11.22: ,„„+„,.,,

We make use of the fact that small, angles are vector a.
p

Then Op = ?^. -i

S£f 1=x

\$ =£*? *£*•**

In averaging over many
■t riu s

&*£%

many traversals, Oj is positive as
e_ — r
1~

often aa it is negative,

Aonendix II .2 . Deri

i greater detail. ?, the avsra C e number of collisions in one
traversal is given fc y

If f is a function o7 X M& N# is the normalized probability
den sit;; forX , then the sean of f is

J"=» fj%j ^fcrj d>K II . 59

anpl ice. 1 c a I cul at e ©\ S 1 = J [©0=) ] * ttfo) d b .

The probability density for b, #J Is proportional %a prow.-
sectional area at the distance b, 2Trb db , to the nuclear
density N," and to the path length D. Therefore the normalized
Wfe) I a Z ir lo D N 2 tt b DM

Therefore

Finally

— i, f[&Cb)] 2 2-rrbDN4b

II. '0

II. 71

(II

Anoendix II.? . Equation 11.26 can be derived by consi dering the
^ ro lections "of the angles A on some chosen plane and deriving a
value forV , the mean square of the resultant P™ -ted angle
Then the unpro jected, true scattering an G ie can be found from MM

result that ± @* = -q> a (^..<<,1) II *

The factor £ comes fror. avera G in G cos 2 , the an G le bein G random,
A "similar relation holds between the actual linear deviation ol

Ch. II

Appendix II ,3

53

Placement on some chosen pSmT'i!S! ^^ct-ion of that dis-

/. emergence

— f^— — I

SlWpsy of entrance ^fc

-g7^ 2 = -x 2

11.73

(plane of paper normal to Incident
direction. Lookin 5 toward source
01 particles)

In a cloud chamber Efe-oissrrartb 1+ i- a +1

scat terms that 'la shownf " ls the ? ro J G ^ed angle of

-^^=^^ used bj danossy

Problem.. Electrons of ensr-py

10° e.v. (from cosmic rays)

pass through a lead nlate 1 G «

thick Calculate the mean square ,

01 the linear deviation of the electron

emergent points from the peo- ""

metrical shadow point on the- last

the patii is almost straight.

|^%«^^iS^^^S?S• ,, f ^-the^sur-
The projected angle of deflexion at ^ n aZ ^ h tn Sle i S £
Projection of linear dissent £ 1" "f**^ - *» ^

DEVIATION »

last
irom the/^sur-

for colli -
sion =£; :

PROJECTION \^"

OP PAT h v ,. ,

*■ -D

1=

& * FNCE TOWARD SOURCE OP

Particles.

**^pHj%%^ S 1§^&, bat C^c^ = X5

^^ff / 3,Mce f* a "d i a r e independent

" ^ "^ 6 2 P P= averse ho. of c.o|[ hS ions

Th^un.^ected linear (dlspW^ht^ ^ , will be ^0.
^ 2 =^D*©* (^0.02 Cffl. USln£ H.31)

54

Arrendix II .4

3h. II

g4&

11.74

A-Q^endix II .4 . Proof that moimnixw cannot be- conserved in hrz
tion of an electron- positron pair by a sir.yle Isolated photon.
By conservati on of energy : L

By oonseri r ation of momentum:

(The equality holds if 3 + and 3~ f/b ° r1 ' ln the same direction)

Fro, n.75 (M*^ #c**4L*/e» + MM?^

Fro, II .74 M" = *fcV * * ^Vzt^lM^ Pf^^ x 2

The radical terra is larper than 2yjo +/ f5./o 2 , so both carrot be
satisfied .

References: Books and articles referred to ;:ore than once in
this chapter are listed.

Ab raham- Be c k e r , The orie der Slectrlzitat , Vol, II ( T eub re r ,

Ed.uards)

Heitlcr, Quantum Theory of Radiation , 2nd ed . (Oxford)

Janossy, Cosmic Ray s (Oxford)

Livingston and Be the , Rev . Mod . Phy s . , £, 245 (1937) (experimental

.nuclear phy s i c s )

Rutherford, Chad wick and Ellis, Rad i at i c n s f ro n Radi oa c t iv e

Substances , ( C a mb rid ge~)

Rossi and Greisen, Rev . Mod . Phys . 13, 249 (1941)

```